### Video Transcript

Find the absolute maximum and
minimum values rounded to two decimal places of the function π of π₯ equals five
π₯π to the negative π₯, given that π₯ is a part of the closed interval zero to
four.

Remember, to find absolute extrema
for our function π of π₯, we follow three steps. We find all critical points in our
closed interval. We then find the values of π of π₯
at these critical points. And then, we check the end points
for absolute extrema. The critical points are the points
where the derivative is equal to zero or does not exist. So, weβre going to find the
derivative of our function and set that equal to zero.

And here, we notice that this
itself is the product of two differentiable functions. So, weβre going to use the product
rule. This says that the derivative of
the product of two differentiable functions π’ and π£ is π’ times dπ£ by dπ₯ plus π£
times dπ’ by dπ₯. So, we let π’ be equal to five π₯
and π£ be equal to π to the negative π₯. Then, dπ’ by dπ₯ is equal to five
and dπ£ by dπ₯ π₯ is equal to negative π to the negative π₯. This means the derivative of our
function is five π₯ times negative π to the negative π₯ plus π to the negative π₯
times five, which we can simplify to five π to the negative π₯ times one minus
six

Letβs set this equal to zero. Now thereβs no way for five π to
the negative π₯ to be equal to zero. So, for the statement five π to
the negative π₯ times one minus π₯ equals zero to be true, we know that one minus π₯
itself must be equal to zero, which means that π₯ equals one is a critical
point. Weβre, therefore, going to evaluate
our function at this critical point and at the end points of the function, so π of
one, π of zero, and π of four.

π of one is five times one times
π to the negative one, which is 1.8393 and so on, or correct to two decimal places
as required 1.84. π of zero is zero. And π of four is five times four
times π to the negative four, which is 0.37 correct to two decimal places. And we can, therefore, say that the
absolute maximum value of our function is 1.84 and the absolute minimum value is
zero.